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Figure 11.22: Level lines and graph of the segmentation function in time step 100 (top row). Then we show graphs of segmentation function after 300 and 800 steps (middle row). In the bottom row we plot the segmented Kanizsa triangle (color slide).

The next examples are related to medical image segmentation. First we process a 2D echocardiography (165 x 175 pixels) with high level of noise and gaps in ventricular and atrium boundaries (see Fig. 11.23).

In Fig. 11.24 we present segmentation of the left atrium. We start with peaklike segmentation function, v = 1, and we use e = 10-2, K = 0.1, t = 0.001, TOL = 10-3, and 8 = 10-5. In the top row of the figure we present the result of segmentation with no presmoothing of the given echocardiography. In such a case 68 time steps, with overall CPU time of 6.54 sec, were needed for threshold 8.

Figure 11.23: Echocardiographic image with high level of noise and gaps.

In the top right we see a graph of the final segmentation function. In the middle row we see its histogram (left) and zoom of the histogram around max(u) (right). By that we take level 0.057 for visualization of the boundary of segmented object (top left). In the bottom row we present the result of segmentation using 5 x 5 convolution mask. Such a result is a bit smoother and 59 time steps (CPU time = 5^65 sec) were used.

For visualization of the segmentation level line in further figures, we use the same strategy as above, i.e. the value of u just below the last peak of histogram (corresponding to upper "flat region") is chosen. In segmentation of the right atrium, presented in Fig. 11.25, we took the same parameters as above and no presmoothing was applied. CPU time for 79 time steps was 7.59 sec. In segmentation of the left and right ventricles, with more destroyed boundaries, we use K = 0^5 and we apply 5 x 5 convolution mask (other parameters were same as above). Moreover, for the left ventricle we use double-peak-like initial function (see Fig. 11.26 (top)) to speed up the process for such highly irregular object. In that case 150 time steps (CPU time = 14.5 sec) were used. For the right ventricle, 67 time steps (CPU time = 6.57 sec) were necessary to get segmentation result, see Fig. 11.27.

In the last example given in Fig. 11.28, we present segmentation of the mammography (165 x 307 pixels). Without presmoothing of the given image and with parameters e = 10j1, K = 04, r = 0^0001, v = 1, TOL = 10-3, and 5 = 10-5 we get the segmentation after 72 time steps. Since there are no big gaps, we take larger e and since the object is small (found in a shorter time) we use smaller time step r.

Figure 11.24: Segmentation level line and graph of the segmentation function for computation without convolution (top row) and histogram of the segmentation function and its zoom (middle row). Segmentation level line and graph of the segmentation function for computation with convolution (bottom row) (color slide).

Figure 11.24: Segmentation level line and graph of the segmentation function for computation without convolution (top row) and histogram of the segmentation function and its zoom (middle row). Segmentation level line and graph of the segmentation function for computation with convolution (bottom row) (color slide).

Figure 11.25: Segmentation level line and graph of the segmentation function for the right atrium (color slide).
Figure 11.26: Initial double-peak segmentation function (top) and segmentation level line and graph of the segmentation function for the left ventricle (color slide).
Figure 11.27: Segmentation level line and graph of the segmentation function for the right ventricle (color slide).

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